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- In how many ways can the letters of the word "ENGINEER" be arranged such that no two E's are together?Option: 1 64,127,253<br
In how many ways can the letters of the word "ENGINEER" be arranged such that no two E's are together?
64,127,253
39,916,656
44,789,589
48,258,963
Answers (1)
To calculate the number of ways the letters of the word "ENGINEER" can be arranged such that no two E's are together, we can treat the three E's as distinct entities (E1, E2, and E3). This allows us to treat the problem as arranging the letters "N, G, I, N, R" along with the three distinct E's.
The word "ENGINEER" has a total of 8 letters, but since the three E's are treated as distinct entities, we have a total of 8 + 3 = 11 objects to arrange.
We can arrange these 11 objects in 11! ways. However, within this arrangement, we have to account for the cases where the E's are together.
Consider the three E's as a single entity. We can arrange these 4 entities ("ENG1, N, G, R") in 4! ways. Within each of these arrangements, the three E's can be rearranged among themselves in 3! ways.
Therefore, the total number of arrangements where no two E's are together is given by:
.
Thus, there are 39,916,656 ways to arrange the letters of the word "ENGINEER" such that no two E's are together.
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